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[ 370 ] 

quodque folum, certa nitri figna pr^ebcre, fed plura 
concurrere debere^ ut de vero nitro produdo dubium 
non relinquatur. 



LI I. AnEffay towards folvlng a Problem in 
the DoSirine of Chances. By the late Rev. 
Mr. Bayes, F. R. S. communicated by Mr. 
Price, in a Letter to John Canton, A. M. 
F. R. S. 

Dear Sir, 

Read Dec. 23, TT Now fend you an eflay which I have 
^763- J^ found among the papers of our de- 
ceafed friend Mr. Bayes, and which, in my opinion, 
has great merit, and well deferves to be preferved. 
Experimental philofophy, you will find, is nearly in- 
terefted in the fubjed of it ; and on this account there 
feems to be particular reafon for thinking that a com- 
munication of it to the Royal Society cannot be im- 
proper. 

He had, you know, the honour of being a mem- 
ber of that illuftrious Society, and was much efteem- 
ed by many in it as a very able mathematician. In an 
introduaion which he has writ to this Eflay, he fays, 
that his defign at firft in thinking on the fubjea: of it 
was, to find out a method by which we might judge 
concerning the probability that an event has to hap- 
pen, in given circumftances, upon fuppofition that we 
know nothing concerning it but that, under the fame 

circum- 



[ 371 ] 

circumftances, it has happened a certain number of 
times, and failed a certain other number of times. 
He adds, that he foon perceived that it would not be 
very difficult to do this, provided fome rule could be 
found according to which we ought to eftimate the 
chance that the probability for the happening of an 
event perfectly unknown, fhould lie between any two 
named degrees of probability, antecedently to any ex- 
periments made about it ; and that it appeared to him 
that the rule muft be to fuppofe the chance the fame 
that it {hould lie between any two equidifferent de- 
grees ; which, if it were allowed, all the reft might 
be eafily calculated in the common method of pro- 
ceeding in the dodtrine of chances. Accordingly, I 
find among his papers a very ingenious folution of this 
problem in this way. But he afterwards confidered, 
that ihcpo/iulate on which he had argued might not 
perhaps be looked upon by all as reafonablej and 
therefore he chofe to lay down in another form the 
propofition in which he thought the folution of the 
problem is contained, and in ajcholium to fubjoin the 
reafons why he thought fo, rather than to take into 
his mathematical reafoning any thing that might ad- 
mit difpute. This, you will obferve, is the method 
which he has purfued in this efTay. 

Every judicious perfon will be fenfible that the 
problem now mentioned is by no means merely a 
curious fpeculation in the dodlrine of chances, but ne- 
ceflary to be folved in order to a fure foundation for all 
our reafonings concerning paft fa<3:s, and what is likely 
to be hereafter. Common fenfe is indeed fufficient 
to fhew us that, from the obfervation of what has in 
former inftances been the confequence of a certain 
5 caufe 



[ 372 1 

caufc or adion, one may make a judgment what is 
likely to be the confequence of it another time, and 
that the larger number of experiments we have to 
fupport a conclulion, fo much the more reafon v/e 
have to take it for granted. But it is certain that we 
cannot determine, at leaft not to any nicety, in what 
degree repeated experiments confirm a conclufion, 
without the particular difcuflion of the beforementi- 
oned problem ; which, therefore, is necefTary to be con- 
fidered by any one who would give a clear account of 
the ftrength of analogical or induSiive reafoniiig \ con- 
cerning, which at prefent, we feem to know little more 
than that it does fometimes in fad convince us, and 
at other times not ; and that, as it is the means of 
cquainting us with many truths, of which otherwife 
we muft have been ignorant ; fo it is, in all proba- 
bility, the fource of many errors, which perhaps 
might in fome meafure be avoided, if the force that 
this fort of reafoning ought to have with us were more 
diftin6tly and clearly underftood, 

Thefe obfervations prove that the problem enquired 
after in this eflay is no Icfs important than it is curi- 
ous. It may be fafely added, I fancy, that it is alfo 
a problem that has never before been folved* Mr. 
De Moivre, indeed, the great improver of this part 
of mathematics, has in his Law^ of chance *, after Ber- 
noulli, and to a greater degree of exadnefs, given 
rules to find the probability there is, that if a very 
great number of trials be made concerning any event, 

* See Mr. De Moivre's DoSfrine of Chances, p. 243, &c. H^ 
has omitted the demonftrations of his rules, but thefe have been 
fince fupplied by Mr, Simpfon at the conclufion of his treatife 
on The Nature and Laws of Chance*, 

the 






[ 373 ] 

the proportion of the number of times it will hap- 
pen, to the number of times it will fail in thofe tri- 
als, ihould differ lefs than by fmall affigned limits 
from the proportion of the probability of its happen- 
ing to the probability of its failing in one fingle trial. 
But I know of no perfon who has fhewn how to de- 
duce the folution of the converfe problem to this ; 
namely, « the number of times an unknown event 
has happened and .failed being given, to find the 
chance that the probability of its happening ihould 
lie fomewhere between any two named degrees of 
probability." What Mr. Dc Moivre has done 
therefore cannot be thought fufficient to make the 
confideration of this point unneceflkry : efpecially, as 
the rules he has given are not pretended to be rigo- 
roufly exadt, except on fuppofition that the number 
of trials made are infinite s from whence it is not ob- 
vious how large the number of trials muft be in or- 
der to make them exadt enough to be depended on 
in praftice, 

Mr. De Moivre calls the problem he has thus folv- 
ed, the hardeft that can be propofed on the fubjeft 
of chance. His folution he has applied to a very 
important purpofe, and thereby fhewn that thofe 
a remuch miftaken who have infinuated that the Doc- 
trine of Chances in mathematics is of trivial confe- 
quence, and cannot have a place in any ferious enqui- 
ry *'. The purpofe I mean is, to mew what reafon 
we have for believing that there are in the conftitutioh 
of things fixt laws according to which events happen, 
and that, therefore, the frame of the world muft be 

* See his DoSrine of Chances, p. 252, &c. 

Vol. LIIL Ccc the 



[ 374 ] 

the eflfed of the wifdom and power of an intelligent 
caufe; and thus to confirm the argument taken from 
final caufes for the exiftence of the Deity. It will be 
eafy to fee that the converfe problem folved in this 
effay is more diredly applicable to this purpofe -, for 
it ihews us, with diftinilnefs and precifion, in every 
cafe of any particular order or recurrency of events, 
what reafon there is to think that fuch recurrency or 
order is derived from ftable caufes or regulations inna-^ 
ture, and not from any of the irregularities of chance. 

The two laft rules in this eflfay are given without 
the dedudions of them. I have chofen to do this 
becaufe thefe deducStions, taking up a good deal of 
room, would fwell the e% too much ; and alfo be- 
caufe thefe rules, though of confiderable ufe, do not 
anfwer the purpofe for which they are given as per^ 
fedly as could be wiftied. They are however 
ready to be produced, if a communication of them 
fhould be thought proper, I have in fome places 
writ fhort notes, and to the whole I have added an 
application of the rules in the effay to fome particu- 
lar cafes, in order to convey a clearer idea of the na- 
ture of the problem, and to fhew how far the folu- 
tion of it has been carried. 

1 am fenfible that your time is fo much taken up 
.hat I cannot r^afonably expeft that yoa fcould mi- 
nutely examine every part of what I now fend you* 
Some of the calculations, particularly in the Appen- 
dix, no one can make without a good deal of labour* 
I have taken fo much care about them, that I believe 
there can be no material error in any of them 5 but 
fliould there be any fuch errors, I am the only per- 
fon who ought to be confidered as anfwerable for 
them. 

Mr. 



[ 375 ] 

Mr. Bayes has thought fit to begin his work with 
a brief demonftration of the general laws of chance. 
His reafon for doing this, as he fays in his introduc- 
tion, was not merely that his reader might not ha?e 
the trouble of fearching elfewhere for the principles 
on which he has argued, but becaufe he did not know 
whither to refer him for a ckar demonftration of 
diem^ He has alfo made an apology for the peculiar 
definition he has given of the word chance or froba^ 
bilky. His defign herein was to cut qW all difpute 
about the meaning of the word^ which in common 
language is ufed in different fenfes by perfons of dif- 
ferent opinions, and according as it is applied to pafi 
ox future fa<5ts. But whatever different lenies it may 
have, all (he obferves) will allow that an exped:ation 
depending on the truth of any paji iz€k, or the hap- 
pening of zny future event, ought to be eftimated fo 
much the more valuable as the fad is more likely to 
be true, or the event more likely to happen. Inftead 
therefore, of die proper fenfe of the word frobabp- 
Utyy he has given that vi^hich all will allow to be its 
proper meafure in every cafe where the word is ufed. 
But it is time to conclude this letter. Experimental 
philofophy is indebted to you for feveral difcoveries 
and improvements i and, therefore, I cannot help 
thinking that there is a peculiar propriety in direft- 
ing to you the following effay and appendix. That 
your enquiries may be rewarded with many further 
fiiccefies, and that you may enjoy every eveiy valuable 
bleffing, is the fincere wiih of. Sir, 

your very humble fervant, 

Newington-Green, ^. ^ t ^ * 

Nov. 10, 1763. Richard Price* 

Ccc 2 SEC- 



[ 376 ] 

P R O B L E M. 

Given the number of times in which an unknown 
event has happened and failed: Required the chance 
that the probability of its happening in a lingle trial 
lies fomewhere between any two degrees of pro- 
bability that can be named. 



D 



SECTION L 

E F I N I T I O N I. Several events are in-^ 
conjtjienty when if one of them happens, none 
of the reft can. 

2. Two events are contrary when one, or other of 
them mufti and both together cannot happen, 

3. An event is faid to fail, when it cannot hap- 
pen s or, which comes to the fame thing, when its con- 
trary has happened. 

4. An event is faid to be determined when it has 
either happened or failed. 

5. The probability of any event is the ratio between 
the value at which an expedation depending on the 
happening of the event ought to be computed, and 
the value of the thing expeded upon it's happening. 

6. By chance I mean the fame as probability. 

7. Events are independent when the happening of 
any one of them does neither increafe nor abate the 
probability of the reft.. 

PROP. r. 

When feveral events are inconliftent the probabili-- 
ty of the happening of one or other of them is the 

fum of the probabilities of each of them. 

Suppofe 



[ 377 ] 

Suppofe there be three fuch events, and which ever 
of them happens I am to receive N, and that the pro- 
bability of the I ft, 2d, and 3d are refpedtively ^, 
l^> ~. Then (by the definition of probability) the va- 
lue of my expedation from the i ft will be a, from 
the 2d ^, and from the 3d ^. Wherefore the value 
of my expeftations from all three will be ^4-^ +- ^. 
But the fum of my expectations from all three is in 
this cafe an expeftation of receiving N upon the hap- 
pening of one or other of them. Wherefore (by de- 
finition 5) the probability of one or other of them is 

bilities of each of them. 

Corollary. If it be certain that one or other 
of the three events muft happen, then a -^ if -\'C 
•= N. For in this cafe all the expectations to- 
gether amounting to a certain expedation of re- 
ceiving N, their values together muft be equal 
to N, And from hence it is plain that the proba- 
bility of an event added to the probability of its fai- 
lure (or of its contrary) is the ratio of equality. For 
thefe are two inconfiftent events, one of which ne- 

ceiTarily happens. Wherefore if the probability of 

P , ^5 ^p 

an event is — that of it's failure will be ■ — rv - . 

PROP. 2. 

If a perfon has an expedation depending on the 
happening of an event, the probability of the event 
is to the probability of its failure as his lofs if it fails to 
his gain if it happens. 

Suppofe a perfon has an expedation of receiving 
N, depending on an event the probability of which 

is 



[378 ] 

• P 

is rr. Then (by definition 5) the value of his ex- 

pedation is P, and therefore if the event fail, he lofes 
that which in value is P ; and if it happens he re- 
ceives N, but his expedation ceafes. His gain there- 
fore is N — ^P. Likewife fince the probability of the 

event is —, that of its failure (by corollary prop, i) 

• N— P ^r^ P . N— P T> . , XT -D • 

IS - ^ ■■-, But rr- IS to -r-T— as F IS to JN — 1 , 1. e, 

N N N 

the probability of the event is to the probability of it's 
failure, as his lofs if it fails to his gain if it happens. 

PROP. 3. 

The probability that two fubfequent events will 
both happen is a ratio compounded of the probabi- 
lity of the I ft, and the probability of the 2d on fup- 
pofition the ift happens. 

Suppofe that, if both events happen, I am to receive 

N, that the probability both will happen is ^ , that 

the I ft will is ~ (and confequently that the ift will 

not is -~-) and that the ad will happen upon fup- 

pofition the ift does is rr-. Then (by definition 5) P 

will be the value of my expedation, which will be- 
come b if the ift happens. Confequently if the ift 
happens, my gain by it is ^ — P, and if it fails my lofs 

is P. Wherefore, by the foregoing propofition, ^ is to 
^, i. e. a is to N— ^ as P is to ^—P. Where- 

fore (componendo inverfe) ^ is to N as P is to k 
But the ratio of P to N is compounded of the ratio 
of P to ^, and that of i to N. Wherefore the 

5 f^^^ 



[ 379 ] 

fame ratio of P to N is compounded of the ratio of 
^ to N and that of b to N, i. e, the probability that 
the two fubfequent events will both happen is com- 
pounded of the probability of the i ft and the proba- 
bility of the 2d on fuppolition the i ft happens. 
Corollary. Hence if of two fubfequent events the 

probability of the ift be ^, and the probability of 
both together be—, then the probability of the 2d 
on fuppolition the ift happens is -. 

PRO P. 4. 

If there be two fubfequent events to be determined 
every day, and each day the probability of the 2d is 

|j and the probability of both |^, and I am to re- 
ceive N if both the events happen the ift day on 

which the 2d does^l fay, according to thefe con- 

p 
ditions, the probability of my obtaining N is j. For 

if not, let the probability of my obtaining N be :|r 

and let ^ be to at as N— -iJ to N. Then fince ^ is the 

probability of my obtaining N (by definition i) x is 
the value of my expedation. And again, becaufe ac- 
cording to the foregoing conditions the ift day I have 
an expeftation of obtaining N depending on the hap- 
pening of both the events together, the probability of 

which is — , the value of this expectation is P. Like- 
wife, if this coincident fhould not happen I have an 
expedation of being reinftated in my former circum- 
ftances, i.e. of receiving that which in value is ^ de- 

pending 



t 380 ] 

pending on the failure of the 2d event the probability 

of which (by cor. prop, i) is -r^ or-, becaufe y is 

to X as N — b to N. Wherefore fince x is the thing 

expeded and - the probability of obtaining it, the 

value of this expedlation is y. But thefe two laft ex- 
pedations together are evidently the fame with my 
original expectation, the value of which is x, and 
therefore P ^y = x. But ^ is to at as N — b is to N. 

Wherefore x is to P as N is to by and ^ (the 

probability of my obtaining N) is 7. 

Cor. Suppofe after the expectation given me in the 
foregoing propofition, and before it is at all known 
whether the ift event has happened or not, I fliould 
find that the 2d event has happened; from hence I 
can only infer that the event is determined on which 
my expectation depended, and have no reafon to 
efteem the value of m.y expectation either greater or 
lefs than it was before. For if I have reafon to think 
it lefs, it would be reafbnable for me to give fomething 
to be reinftated in my former circumftances, and 
this over and over again as often as I fhould be in- 
formed that the 2d event had happened, which is evi- 
dently abfurd* And the like abfurdity plainly follows 
if you fay I ought to fet a greater value on my expec- 
tation than before, for then it would be reafonable for 
me to refufe fomething if offered me upon condition 
I would relinquifh it, and be reinftated in my former 
circumftances ; and this likewife over and over again 
as often as (nothing being known concerning the ifl 
event) it fhould appear that the 2d had happened. 
Notwithfl^nding therefore this difcovery that the 2d 

event 



[ 38i ] 

event has happened, my expedation ought to be 
efteemed the fame in value as before, i. e. ^, 
and confequently the probability of my obtaining 

X P 

N is (by definition 5) flill |;t or t-"*. But after this 

difcovery the probability of my obtaining N is the pro- 
bability that the ift of two fubfequent events has hap- 
pened upon the fuppoiition that the 2d has, whofe pro- 
babilities were as before fpecified. But the probability 
that an event has happened is the fame as the proba- 
bility I have to gueis right if I guefs it has happened. 
Wherefore the following propofition is evident. 

PRO P. 5: 

If there be two fubfequent events, the probability 

of the 2d i-a„d the probability of both togeAer \, 

and it being ift difcovered that the 2d event has hap- 
pened, from hence I guefs that the ift event has al- 

fo happened, the probability I am in the rightis t ''• 

PROP. 

* What is here faid may perhaps be a little illuftrated by con* 
fidering that all that can be loft by the happening of the 2d event 
is the chance I fhould have had of being reinftated in my former 
circumftances, if the event on which my expeftation depended had 
been determined in the manner exprefled in the propofition. But 
this chance is always as much againji me as it isj^r me. If the 
I ft event happens, it is againji me, and equal to the chance for 
the 2d event's failing* If the ift event does not happen, it is 
for me, and equal alfo to the chance for the 2d event's failing. 
The lofs of it, therefore, can be no difadvantage. 

f What is proved by Mr. Hayes in this and the preceding pro- 
pofition is the fame with the anfwer to the following queftion. 
What is the probability that a certain event, when it happens, will 

Vol. LIIL D d d be 



[382 ] 



P R O P. 6. 

The probability that feveral independent events 
ihall all happen is a ratio compounded of the proba- 
bilities of each. 

For from the nature of independent events, the 
probability that any one happens is not altered by the 
happening or failing of any of the reft, and confe- 
quently the probability that the ad event happens on 
luppofition the ift does is the fame with its original 
probability ; but the probability that any two events 
happen is a ratio compounded of the probability of the 
ift event, and the probability of the 2d on fuppofition 
the I ft happens by prop. 3. Wherefore the probability 
that any two independent events both happen is a ra- 
tio compounded of the probability of the i ft and the 
probability of the 2d. And in like manner confidering 
the ift and 2d event together as one event 5 the proba- 
bility that three independent events all happen is a ratio 
compounded of the probability that the two i ft both 
happen and the probability of the 3d. And thus you 

be accompanied with another to be determined at the fame time ? 
In this cafe, as one of the events is given, nothing can be due 
for the expectation of it ; and, confequently, thq value of an ex- 
peftation depending on the happening of both events muft be the 
fame with the value of an expedlation depending on the happen- 
ing of one of them. In other words ; the probability that, when 
one of two events happens, the other will, is the fame with the 
probability of this other. Call x then the probability of this 

other, and if - be the probability of the given event, and L 

^ p b i> ^ 

the probability of both, becaufe 7; = - X .v* a* r: 4 r: the pro- 

*■ J si si b 

bability mentioned in thefe propofitions. 

may 



[ 383 ] 

may proceed If there be ever fo many fuch events j 
from whence the propofition is manifeft. 

Cor. I. If there be feveral independent events, the 
probability that the ift happens the 2d fails, the 3d 
fails and the 4th happens, &c. is a ratio compound- 
ed of the probability of the ift, and the probability 
of the failure of the 2d, and the probability of the 
failure of the 3d, and the probability of the 4th, &c. 
For the failure of an event may always be coniidered 
as the happening of its contrary. 

Cor. 2. If there be feveral independent events, and 
the probability of each one be ^, and that of its fail- 
ing be ^, the probability that the i ft happens and the 
2d fails, and the 3d fails and the 4th happens, &c. 
will hQ abba^ &c. For, according to the algebraic 
way of notation, if ^denote any ratio and ^another, 
abba A^nott^ the ratio compounded of the ratios 
a^ b^ by a. This corollary therefore is only a particular 
cafe of the foregoing. 

Definition. If in confequence of certain data 
there arifes a probability that a certain event fhould 
happen, its happening pr failing, in confequence 
of thefe data, I call it's happening or failing in 
the I ft trial. And if the fame data be again re- 
repeated, the happening or failing of the event in 
confequence of thetn 1 call its happening or failing 
in the 2d trial j and fo on as often as the iame data 
are repeated. And hence it is manifeft that the hap- 
pening or failing of the fame event in fo many difFe- 
trials, is in reality the happening or failing of fo 
many diftind independent events exadiy fimilar to 
each other. 



Ddd 2 PROP. 



[ 384 ] 



K U Jr. y. 

If the probability of an event be a^ and that of its 
failure be b in each fingle trial, the probability of its 
happening p times, and failing q times xnp^q trials 
is E db^ if E be the coefficient of the term in which 

occurs d b* when the binomial a -|-^j ^"*"^ is ex- 
panded. 

For the happening or failing of an event in differ- 
ent trials are fo many independent events. Where- 
fore (by cor. 2. prop. 6.) the probability that the event 
happens the ift trial, fails the 2d and 3d, and hap- 
pens the 4th, fails the 5th, &c. (thus happening and 
failing till the number of times it happens be p and 
the number it fails be q) is abbab &cq. till the 
number of ^'s be^ and the number of Fs be ^, that 
is; 'tis d b\ In like manner if you confider the event 
as happening p times and failing q times in any other 
particular order, the probability for it is d b^\ but 
the number of difRrent orders according to which an 
event may happen or fail, io as in all to happen ^ 
times and fail q^ mp \ q trials is equal to the num- 
ber of permutations that aaaa bbb admit of when 
the number of d% is /, and the number of ^'s is q. 
And this number is equal to E, the coefficient of the 

term in which occurs d b^ when a^ ^^Hs ex- 
panded. The event therefore may happen / times 
and fail f in ^ -|- 5^ trials E different ways and no 
more, and its happening and failing thefe feveral dif- 
ferent ways are fo many inconfiflent events, the pro- 
bability for each of which is a^ b^^ and therefore by 

prop. 



[ 38s ] 

prop* I. the probability that fome way or other it 
happens p times and fails q times m p ^q trials is 
E a^ bK 

S E C T I O N 11. 

Poftulate. I. I Suppofe the fquare table or plane 
A B C D to be fo made and levelled, that if either 
of the balls o or W be thrown upon it, there fhall 
he the fame probability that it refts upon any one 
equal part of the plane as another, and that it muft 
ueceflarily reft fomewhere upon it. 

2. I fuppofe that the ball W fliall be ift thrown, 
and through the point where it refts a line os fhall be 
drawn parallel to AD, and meeting CD and AB in 
J and <?; and that afterwards the ball O fliall be 
thrown / 4- ^ or ;^ times, and that its refting between 
AD and os after a fingle throw be called the hap- 
pening of the event M in a fingle trial* Thefe things 
fuppofed, 

Lem. I. The prQba-.n F » s H I K • L x\ 

bility that the point 
will fall between any 
two pints in the line 
A B is the ratio of the 
diftance between the 
two points to the whole 
line AB. 

Let any two points 
be named, as f and k 
in the line AB, and 8 
through them parallel 
to A D draw fF^ b L 
meeting C D in F and 
L. Then if the red- 
angles Cj^ F 4 L A are com^ 




[ 386 ] 

commenfurable to each other, they may each be di- 
vided into the fame equal parts, which being done, 
and the ball W thrown, the probability it will reft 
fomewhere upon any number of thefe equal parts 
will be the fum of the probabilities it has to reft upon 
each one of them, becaufe its refting upon any differ- 
ent parts of the plane AC are fo many inconfiftent 
events ^ and this fum, becaufe the probability it Ihould 
reft upon any one equal part as another is the fame, is 
the probability it fliould reft upon any one equal part 
multiplied by the number of parts. Confequently, the 
probability there is that the ball W fhould reft fome- 
where upon F^ is the probability it has to reft upon one 
equal part multiplied by the numberof equal parts in Yh 
and the probability it rells fomewhere upon 6/ or LA, 
i.e. that it dont reft upon F^ (becaufe it muft reft fome- 
where upon A C) is the probability it refts upon one 
equal part multiplied by the number of equal parts in 
C/, L A taken together. Wherefore, the probability 
it refts upon Fi is to the probability it dont as the 
number of equal parts in F^ is to the number of 
equal parts in CJ] L A together, or as ^ b to Cf, 
LA together, or as/i to B/ A^ together. Where- 
fore the probability it reft upon F^ is to the proba- 
bility it dont as fb to B/, Kb together. And (com- 
ponendo imerfe) the probability it refts upon F^ is to 
the probability it refts upon ¥b added to the proba- 
bility it dont, as fb to A B, or as the ratio oi fb to 
AB to the ratio of AB to AB. But the probabi- 
lity of any event added to the probability of its failure 
is the ratio of equality; wherefore, the probability it 
reft upon F ^ is to the ratio of equality as the ratio of 
Jb to AB totheratioof AB to AB, or the ratio 
of equality j and therefore the probability it reft upon 

¥b 



[ 387 3 

F^ is the ratio of/3 to AB, But e^ h)fothefi ac- 
cording as the bail W falls upon F 3 or not the 
points will lie between /and b or not, and there- 
fore the probability the points will lie between/ and 
b is the ratio of /I to AB. 

Again s if the rectangles C/ F^ LA are not 
commenfurable, yet the laft mentioned probability 
can be neither greater nor lefs than the ratio of/3 to 
A B y for, if it be lefs, let it be the ratio of /^ to AB, 
and upon the line/3 take the points p and ty h 
that p t fhall be greater than fc^ and the three lines 
B^, pfy tK commenfurable (which it is evident may 
be always done by dividing A B into equal parts lels 
than half <r3, and taking p and / the neareft points 
of divifion to/and ^ that lie upon /3). Then 
becaufe Bp^ pt^ t A are commenfurable, fo arc the 
recSangles C/, D /, and that upon p f compleating 
the fquare AB. Wherefore, by what has been faid, 
the probability that the point will lie between p and 
if is the ratio of /if to AB. But if it lies between p 
and t it muft lie between / and b. Wherefore, the 
probability it fliould lie between / and b cannot be 
lefs than the ratio of p i to A B, and therefore muft 
be greater than the ratio of fc to AB (fince pt h 
greater than fc). And after the fame manner you 
may prove that the forementioned probability cannot 
be greater than the ratio of /3 to A B, it mull there- 
fore be the fame. 

Lem. 2. The ball W having been thrown, and 
the line os drawn, the probability of the event M 
in a fingle trial is the ratio of Ae? to AB. 

For, in the fame manner as in the foregoing lem- 
ma, .he probabili^ .bat .h= baU . being .h?cwf ftall 

refc 



[ 388 ] 

reft fomewhere upon D^ or between AD and so is 
is the ratio of A ^ to A B. But the refting of the 
ball between A D and s o after a fingle throw is 
the happening of the event M in a llngle trial. 
Wherefore the lemma is manifeft. 

PROP. 8. 

If upon BA you ere£t the figure BghikmK 
whofe property is this, that (the bafe B A being di- 
vided into any two parts, as A^, and 15 b and at the 
point of divifion b a perpendicular being ereded and 
terminated by the figure in m 5 and y^ x^ r repre- 
fenting refpe<5tively the ratio of bniy A A, and BA to 
A B, and E being the the coefficient of the term in 

which occurs a^ b'^ when the binomial aJ^tY'^^ is 

expanded) yz=:'Ex^ r ^. I fay that before the ball W 
is thrown, the probability the point fhould fall be- 
tween / and A, any two points named in the line 
AB, and withall that the event M fhould happen p 
times and fail q in p -^ q trials, is the ratio of 
fghikmby the part of the figure BghikmA in- 
tercepted between the perpendiculars y^, bm raifed 
upon the line A B, to C A the fquare upon A B. 

DEMONSTRATION. 

For If not 5 ift let it be the ratio of D a figure 
greater than fgbikmb to CA, and through the 
points Cy dj c draw perpendiculars to fb meeting the 
curve Kmigl^ in h,i,k*, the point d being fo 
placed that di (hall be the longeft of the perpendi- 
^ culars 



[389] 

culars terminated by the line Jb^ and the curve 
AmtgB ; and the points e, d, c being fo many and 
fo placed that the reftangles, bk^ c /, e i, fb taken 
together fhall differ lefs from Jghikmb than D 
does y all which may be eafily done by the help of the 
equation of the curve, and the difference between D 
and the figure Jghikmb given. Then fince di is 
the longeft of the perpendicular ordinates that infift 
upon fbj the reft will gradually decreafe as they are 
farther and farther from it on each fide, as appears 
from the conftrudion of the figure, and confequently 
^ ^ is greater than g/ or any other ordinate that in- 
fifts upon e/^. 

Now if Ao were equal to A^, then by lem. 2. 
the probability of the event M in a fingle trial would 
be the ratio of A ^ to A B, and confequently by cor. 
Prop. I. the probability of it's failure would be the 
ratio of B ^ to A B. Wherefore, if x and r be the 
two forementioned ratios refpeftively, by Prop. 7. the 
probability of the event M happening p times and 
failing q in p ^q trials would be Ex^ r^. But x 
and r being refpedlively the ratios of A ^ to A B 
and B^ to AB, if ;^ is the ratio of eh to AB, then, 
by conftrudtion of the figure A/B, y z=zEx^ r^. 
Wherefore, if A ^ were equal to A ^ the probability 
of the event M happening / times and failing q in 
p-\-q trials would be y^ or the ratio of eh to A B, 
And if Ao were equal to A/, or were any mean be- 
tween Ae and Af^ the laft mentioned probability 
for the fame reafons would be the ratio of y^ or fome 
other of the ordinates infifting upon ef^ to A B. But 
eh is the greateft of all the ordinates that infift upon 
ef. Wherefore, upon fuppofition the point fhould lie 
Vol. LIII. Eee any 



[ 390 ] 

any where between f and e^ the probability that the 
event M happens p times and fails qmp^q tri- 
als can't be greater than the ra,tio of ^^ to AB. 
There then being thefe two fubfcquent events, the 
ift that the point o will lie between rand/, the 
2d that the event M will happen p times and fail q 
in p 4" f trials, and the probability of the ift (by 
lemma ift) is the ratio of ef to AB, and upon fup- 
pofition the Ift happens, by what has been now 
proved, the probability of the 2d cannot be greater 
than the ratio oi eh to A B, it evidently follows (from 
Prop. 3.) that the probability both together will hap- 
pen cannot be greater than the ratio compounded of 
that of f/ to AB and that of eh to AB, which 
compound ratio is the ratio oi fh to C A Where-^ 
fore, the probability that the point will lie between 
/' and ey and the event M happen p times and fail 
i is no. greater than the ratio of /A .o C A. And 
in like, manner the probability the point will lie be- 
tween e and ^, and the event M happen and fail as 
before, cannot be greater than the ratio of ei to C A. 
And pgain, the probability the point will lie betveeen 
d and Cy and the event M happen and fail as before, 
cannot be greater than the ratio of ^/ to C A. And 
laftly, the probability that the point will lie between 
c and by and the event M happen and fail as before, 
cannot be greater than the ratio of ^ ^ to C A. Add 
now all thefe feveral probabilities together, and their 
fum (by Prop, i . ) will be the probability that the point 
will lie fomewhere between f and b^ and the event 
M happen p times and fail q m p J^ q trials. Add 
likewife the correfpondent ratios together, and their 
fum will be the ratio of the fum of the antecedent's 

to 



[ 391 ] 

to their common confequent, i, e. the ratio of //&, 
eiy ci^ hk together to CA; which ratio is lefs 
than that of D to C A, becaufc D is greater 
than fhy e /, ci^ bk together. And therefore, the 
probability that the point o will lie between y* and 4 
and withal that the event M will happen p times 
and fail q in p A- ^ trials, is lefs th^n the ratio of 
D to C A ; but it was fuppofed the fame which is 
abfurd. And in like manner, by infcribing reftanglcs 
within the figure, as eg^ dhy dk^ cm^ you may prove 
that the laft mentioned probability is greater than the 
ratio of any figure lefs than fghikmb to C A, 

Wherefore, that probability muft be the ratio of 
fghikmb to CA. 

Cor. Before the ball W is thrown the probability 
that the point o will lie fomewhere between A and B, 
or fomewhere upon the line A B, and withal that the 
event M will happen p times, and fail q in / 4" ? 
trials is the ratio of the whole figure A/B to C A. 
But it is certain that the point o will lie fomewhere 
upon A B. Wherefore, before the ball W is thrown 
the probability the event M will happen p times and 
fail q m p '\- q trials is the ratio of A / B to C A. 

PROP. 9. 

If before any thing is difcovered concerning the 
place of the point (?, it fhould appear that the event 
M had happened p times and failed qvixp^q trials, 
and from hence I guefs that the point lies between 
any two points in the line A B, as /and b^ and con- 
fequently that the probability of the event M in a An- 
gle trial was Ibmewhere between the ratio of A b to 
A B and that of A /to A B : the probability I am in 

E e e 2 the 



[ 392 ] 

the right Is the ratio of that part of the figure A /B 
defcribed as before which is intercepted between 
perpendiculars ereded upon A B at the points f 
and b^ to the whole figure A / B. 

For, there being thefe two fubfequent events, 
the firft that the point o will lie between f and b \ 
the fecond that the event M fhould happen / times 
and fail^y in p -|- q trials s and (by cor. prop. 8.) the 
original* probability of the fecond is the ratio of 
A / B to C A, and (by prop. 8.) the probability of 
both is the ratio oi f g h imb to C A 5 wherefore 
(by prop. 5) it being firft difcovered that the fecond 
has happened, and from hence I guefs that the 
firft has happened alfo, the probability I am in 
the right is the ratio of fghimb to A/B, the 
point which was to be proved. 

Cor. The fame things fuppofed, if I guefs that 
the probability of the event M lies fome where be- 
tween and the ratio of A ^ to A B, my chance 
to be in the right is the ratio oi K b m to A / B. 

Scholium- 

From the preceding propofition it is plain, that 
in the cafe of fuch an event as I there call M, from 
the number of times it happens and fails in a cer- 
tain number of trials, witiiout knowing any thing 
more concerning it, one may give a guefs where- 
abouts it's probability is, and, by the ufual methods 
computing the magnitudes of the areas there menti- 
oned, fee the chance that the guefs is right. And that 
the fame rule is the proper one to be ufed in the cafe 
of an event concerning the probability of which 

we 



[ 393 ] 

we abfolutely know nothing antecedently to any 
trials made concerning it, feems to appear from the 
follov/ing confideration ; viz. that concerning fuch 
an event I have no reafon to think that, in a certain 
number of trials, it fhould rather happen any one 
pofTible number of times than another. For, on 
this account, I may juftly reafon concerning it as if 
its probability had been at iirft unfixed, and then 
determined in fuch a manner as to give me no reafon 
to think that, in a certain number of trials, it fhould 
rather happen any one poflible number of times 
than another. But this is exadtly the cafe of the 
event M, For before the ball W is thrown, which 
determines it's probability in a fingle trial, (by cor. 
prop. 8.) the probability it has to happen p times 
and fail qinp '\- qox n trials is the ratio of A / B to 
C A, which ratio is the fame when p '\' q ox n\% 
given, whatever number ^ is ; as will appear by 
computing the magnitude of A / B by the method 
* of fluxions. And confequently before the place 
of the point o is difcovered or the number of times 
the event Mhas happened in n trials, I can have no 
reafon to think it ihould rather happen one pof-. 
fible number of times than another. 

In what follows therefore I Ihall take for granted 
that the rule given concerning the event M in 
prop. 9. is alfo the rule to be ufed in relation to any 
event concerning the probability of which nothing 

* It will be proved prefently in art. 4. by computing in the 
method here mentioned that A / B contrafted in the ratio of E 

to 1 is to C A as I to wf I xE : from whence it plainly follows 
that, antecedently to this contraction, A / B muft be to C A in 
the ratio of i to « + 1, which is a conftant ratio when n is given, 
whatever p is, 

at 



[ 394 ] 

?it all Is known antecedently to any trials made or ob- 
ferved concerning it. And fuch an event 1 mall call 
an unknown event. 

Cor. Hence, by fuppofing the ordinates in the fi- 
gure A/B to be contradled in the ratio of E to one, 
which makes no alteration in the proportion of the 
parts of the figure intercepted between them, and 
applying what is faid of the event M to an unknown 
event, we have the following propofition, which gives 
the rules for finding the probability of an event from 
the number of times it adually h?ippens and fails, 

PROP, 10. 

If a figure be defcribed upon any bafe AH (Vid. 

Fig.) having for it's equation j)? m^^ r^ ; where )\ 
x^ r are relpecftively the ratios of an ordinate of the 
figure infifling on the bafe at right angles, of the 
fegment of the bafe intercepted between the ordinate 
and A the beginning of the bafe, and of the other 
fegment of the bafe lying between che ordinate and 
the point H, to the bafe as their common confequent, 
1 fay then that if an unknown event has happened 
p times and failed q mp -^q trials, and in the bafe 
AH taking any two pomts as jT and t you eredt the 
ordinates fc^ tF at right angles with it, the chance 
that the probability of the event lies fomewhere be- 
tween the ratio of Af to A H and that of A ^ to 
A H, is the ratio of tFCf, that part of the before- 
defcribed figure which is intercepted between the two 
ordinates, to A C F H the whole figure infifting on 
the bafe AH. 

This is evident from prop. 9. and the remarks made 
in the foregoing fcholium and corollary. 

5 Now 



[395 ] 

Now, In order to 
reduce the forego- 
ing rule to pradice, 
we muft iind the 
value of the area 
of the figure de- 
fcribed and die fe- 
veral parts of it fe- tt 

parated, by ordi- " 

nates perpendicu- 
lar to its bafe. For 

which purpofe, fuppofe A H rrr i and H O the 
fquare upon A H likewife ■=: i , and Cf will be z=y, 
and Af-=: x, and Hfz=: r, becaufe y^ ^ and r denote 
the ratios of Cy, A J] and II J^ refpedively to AH. 

And by the equation of the curve jy =Ar^r^and (be- 
caufe Af -^/H = A H) r -|- AT = I. Wherefore 




y z=zx^ X i-M ^ 



X' 



qx 



p -^ % 



qx q-i X X 



9. 



X §'- 1 X q-'2. XX -\- &c. Now the abfcifle being 
X and the ordinate x the correlpondent area is x 



p*^ J 



(by prop. 10. caf. i. Quadrat. Newt.) ^ and the ordi- 
nate being g x the area is j' ^v y and iji like man- 

* Tis very evident here, without having recourfe to Sir Ifaac 
Newton, that the fluxion of the area AC/ being yxzzx^x — 

qx X + qxq^i X X &c. the fluent or area itfclf is x^ 

-. q ><>__ + q X £2? ^ ^^ ^ &C. 

ner 



ner of the reft. Wherefore, the abfciiTe being x and 

P P+i 
the ordinate J or ^ ^gx -^ &cc. the correfpondent 

^ + 1 p + 2 ^4-3 

area is x ^gxx -\- q K g-t X x - ^ x q-i X 

P^i p+2 2 p-h3 2 

^-2 X ^ + &c* Wherefore, if x == Ay' = A^, 

3 P + i- A.H 

andjy = C/=C^ then ACfz= ACf =zx 

AH HO >+i 

^+2 /' + 3 

-^q y. X ■^-qxg-ix X — &c. 

^^^,^,g,g„p^0 MBwnpWHHH* 11,1.1 M l > m '»mtmmm* m 

From which equation, if ^^ be a fmall number, it is 
eafy to find the value of the ratio of A C/ to HO. 
and in like manner as that was found out, it will ap- 

q+ I 

pear that the ratio of HCf to HO is r — ^ x 

q+l 
q+t ?+3 f+4 

f+2 2 f+3 ^ 3 ^ + 4 

which feries will confift of few terms and therefore 
is to be ufcd when p is fmalL 

2. The fame things fuppofed as before, the ratio of 

^+1 ^+2 

AC/toHOis^ r^^A-qy.x r^ +?X 

*^ •Hi«HaiHMk-w« ' ' ■ -I* ■Ill |ii1 1 I «■ ■■■ «i I ■ 

^+1 J»+I ^+2 ^+1 

j? + 3 ^+4 

g^i: x ^ r^ ^ + y X j^-i X y-a x ^ ^^''^ + 

JTT F+3 f+i /»+^ /^+3 ^+4- 

&c. 



[ 397 ] 

ice. J^x X 9' X J--! X &c. X t where « =s 

^gl0mmmmmfi» mmmmmrmmtm iMMMwriHMHMMiw nrntH^ 

n+i ^+1 p-jrZ » 

f + t 

p J^q. For this feries is the fame with :>c ^•— q X 

p+2 p+i 

X &c. fet down in Art. ift. as the value of the 



■M 



p'jr2 

ratio of A Cf to H O j as will eafily be fern by put- 
ting in the former inftead of r its value i-x, and 
expanding the terms and ordering them according to 
the powers of x. Or, more readily, by comparing the 
fluxions of the two feries, and in the former inftead 
of r fubftituting ^x^. 

* The fluxion of the firft feries i$ pc r x + gx t^ r + 

P-^^q^t P + ^a^M P + ^q^2 

qx r X + q X q-'l X x r r- + q X f-*I XX r j^ 

jf> + ' /+^ i^+* ^+* ^Hh* 

+ q X f-i X f-3 X X r^**? &c. or, fubftituting - i for r^ 

X r X '-^ a X r^ x + qx r^ ^— fX y— i x 

p+% p+% ^^ 

^ r^ "^x + q X q^t X X r^^ x &c. which, as all the 



mmmmmmmm 



P + 2 p + I p + 2 

terms after the firft deftroy one another, is eq ual to x^ r^ x r=, 

x^ X i— %y|^ ;v = ^i^ ;c X I -^qx-^-qX q — l ^"^ &c. == A'^ ^ — 

^+1 p + 2 2 t /• • 

f y ;^ + f X f-i *• ^ &c, =: the fluxion of the latter fcnes 

P+ 1 * p + S3t 

or of ^ — f X ^ &c* The two feries therefore arc 

P+I P+2 

the fame. 

Vol. LIIL Fff 3* ^^ 



[ 39B ] 

3- In like manner, the ratio of HCf to HO i« 

r x^ -\- p X r x^'^ +/^ X p-^i X ^ ^___ ^^ 

&c. 

4. If E be the coefficient of that term of the bi- 

nomical a -}- ^[/' + ^ expanded in which occurs ^/^ ^^> 
the ratio of the whole figure ACFH to HO is 

TTi ^ "e' ^ being =/) -f-j^. For, when Ayi=: A H 

x^= i^ r=io. Wherefore, all the terms of the fe- 
lies fet down in Art. 2. as expreffing the ratio of 
A Cf to H O will vanifh except the laft, and that 

becomes -tt X ;tt X S X &c. X J . But E 
being the coefficient of that term in the binomial 

a -\- b^ expanded in which occurs a^ b^ is equal to 

t±i X ^i^ X &c. X ^. And,becaufe Ay isfup- 

pofed to become = AH, A C/= A C H. From 
whence this article is plain. 

5, The ratio of AC/ to the whole figure ACFH 



'mmm^mmi'mmi^nmmmitm 



is (by Art. i. and 4.) ;^ -|- i x E x ^ — J' X 
X 4" y >^ ?-^ X ^ ^^' ^^^ ^f:) ^^ ^ expreffes 

>+2" '^T" A+ 3 

the ratio of A/ to AH, X fhould expreis the ratia 
of A^ to AHi the ratio of AF/ to ACFH 

would be «-^i X E X X — ^X +$'X?-i 
X X — &c. and confequ&ntly the ratio of /FC/ 



to ACFH is ;^-j- 1 X E X into the difference 

between 



[ 399 ] 

betwetn the two ferks. Compares this with jprop, io# 
and we ihall have the following pradical mlc. 

RTT T 17 T 
\J Li ill I. 

If nothing is known concerning an event but that 
it has happened ptimts and failed qinpJ^govn trials, 
and from hence I guefs that the probability of its 
happening in a fingle trial lies fomewhere between 
any two degrees of probability as X and ;.,jhe 

chance I am in the right in my guefs is n'\^i 

X Ex into the difference between the feriesX^''"^ 
p+2 p + 3 p+i 

4-f X ?-r X X — &c. and the 




MlMaali^ mmmmm0mffm. 



P+i p + ^ P+3 

feriep x -^ f ^ +f X f-i X x — &c. E 

^+1 p + 2 2 j> + 3 

being the coefficient of a^ b^ when ^-|- bX is expanded. 

This is the proper rule to be ufed when j' is a fmall 
number ; but if q is large and p fmall, change every 
where in the feries here fet down p into q and q into/ 
and X into r or i-at, and X into R =: i-Xj which 
will not make any alteration in the difference between 
the two feriefes. 

Thus far Mr. Bayes*s effay. 

With refped to the rule here given, it is further 
to be obferved, that when both / and q are very large 
numbers, it will not be fKjflible to apply it to practice 
on account of the multitade of terms which the fe- 
riefes in it will contain. Mr. Bayes, therefore, by 

F f f 2 an 



[ 400 ] 

an Inveftigation which it would be too tedious to give 
here, has deduced from this rule another, which is as 
follows. 

RULE £• 

If nothing is known concerning an event but that 
it has happened p times and failed y in ^ + q or n 
trials, and from hence I guefs that the probabihty of 

its happening in a fingle trial lies between ^ 4- 2J and 

«k mm 

t — 2; J if /»* ■=— a = *, b=i 1, E the coefficient 

» pq n n 

of the term in which occurs a^ h^ when a -^A] is 
expanded, and 2 = ^^ — - X —;==:=• X E 12^ i^ x 

by the fenes mz -— 1 X -— - — -^ ^ - ^^ 

X 1 X — ^ X — X — &c. 

my chance to be in the right is greater than 

■■ 2 X ^ 

I 4- 2t E ^^ ^s' ^- 2 E ^^ ^^ * and lefs than 

2 s « 



1-2 E ii/^f — 2 E ^i^ if. And if / = f my chance 



is 2 2 exadly. 



« 



* In Mr. Bayes's manufcript this chance is made to be gieater 
,han — .i^^ and lefs than ?#^^. The third term 

in the two dlvifors, as I have given them, being omitted. But 
this being evidently owing to a fmall overfight in the dedui3:ipn 
of this rule, which I have reafon to think Mr. Bayes had himfelf 
difcovered, I have ventured to correal his copy^ and to give the 
Jule as I am fatisficd it ought to be given. 

In 



[ 401 ] 

In order to render this rule fit for ufe In all cafes 
it is only neceflary to know how to find within fuffi- 
cient nearnefs the value o£ E af h and alfo of the 



m^ %' 



feries m z &c ^. With refpeffc to the former 

Mr, Bayes has proved that, fuppofing K to fignify the 
ratio of the quadrantal arc to it's radius, E a^ if will 

be equal to ^;-y==— r x by the ratio whofe hyperbo-- 

^ 12 n /* ? 300 « p 

' ' ' " I — I I I I I I mm — ■———It 

Jt , I II I I r I 

£^ > 1260 ^ ^ p" p 1680 ^ «» ""p ■" 

T + 7-00 X -7 — TT •— - -r &c. where the nume- 

ral coefficients may be found in the following man- 
ner. Call them A, B, C, D, E, &c. Then A r=r 



2. 2. 3 3.4 2* 4- 5 3 2. 6* 7 

loB + A jQ I ^^ 3sC+2iB+A g _ I 

5 * 2.8.9 7 * 2. 10 .11 

126 C + 84D -F 36 B + A p 1^ 

9 2. 12. 13 ""^ 

* A very few terms of this feries will generally give the hyper- 
bolic logarithm to a fuiEcient degree of exaSnefs. A fimilar fe- 
ries has been given by Mr. De Moivre, Mr. Simpfon and other 
eminent mathematicians in an exprefEon for the fum of th^ lo* 
garithms of the numbers 1, 2, 3, 4, 5 to x^ which fum they 

have affer ted to be equal to | log. r + a* 4- | x log» x — x + 
-TTx — -i^x + "ttVo/ &c. c denoting the circumference of a 
circle whofe radius is unity. But Mr. Bayes^ in a preceding pa- 
per in this volume, has demonftrated that, though this expreffion 
will very nearly approach to the value of this fum when o»ly a 
proper number of the firft terms is taken, the whole feries cannot 
exprefs any quantity at all, becaufe, let x be what it will, there 
will be always a part of the feries where it will begin to diverge. 
This obfervation, though it does not much afFe£l the ufe of this 
feries, feems well worth the noticeof mathematicians, 462 



t 4*^2 ] 
^^.±J12£±l^jA±Il±tJt &c. where the co- 

efficients of B, C, D, E, F, &c. in the values of 
D, E, F, &c. are the 2, 3, 4, &c. higheft coeffid- 

cnts in a +^| > ^ + ^I j a -^ ^\ \ &c. expanded; 
affixing in every particular value the leaft of thefe 
eoefficents to B, the next in magnitude to the fur- 
theft letter from B, the next to C, the next to the 
furtheft but one, the next to D, the next to the fur- 
theft but two, and £0 on *• 

With refpeft to the value of the feries mz — 

f-L 4. 2li )^ ^IS. &c. he has obferved that it may be 

calculated direcftly when mz is lefs than i, or even 
not greater than VJ: but when m z is much larger 
it becomes imprafticable to do this j in which cafe he 
fhews a way of eafily finding two values of it very 
nearly equal between which it's true value muft lie. 

The theorem h? gives for this purpofe is as fol- 
lows. 

Let K, as before, ftand for the ratio of the qua- 
drantal arc to its radius, and H for the ratio whofe 

hyperbolic logarithm is ^ ~ |l^ + ^^ ~ 
^xF~ &c. Then the feries ^^ — —^&c, will be 

greater or lefs than the feries '-1— x «-=--—--— x 



.1^ 



« 1 ^ - . I— « 



zmz «+2 « -j- 4 X 4 w^ 2^ 

^ This method of finding thefe coefficients I have deduced 
from the demonftratign of the third lemma at the end of Mr. 
Simpfon's Trcatife on the Nature and Laws of Chance. 



.2 W2* Z* 



3«' 



[ 403 ] 

T + 3 TZZ2^p\% -]- 4, 

2^5 x^'* _____ n 



X ^ - ^ ^ . . J ., ; --X 



— &c. continued to any number of terms, accord- 
ing as the laft term has a pofitive or a negative fign 
before it. 

From fubftituting thefe values of Ea^ if and m z 



m^ z^ , n — 2 m^ z^ 



4- -. — X &c. in the 2d rule arifes a 

3 * 2w 5 

3d rulq, which is the rule to be ufed when mz is of 
fome confiderabie magnitude. 

RULE 3. 

If nothing is known of an event but that it has 
happened f> times and failed q in ^ -f y or n trials^ 
and from hence I judge that the probability of it's 

happening in a fingle trial lies between - -{- z and 
^ -~ z my chance to be right is greater than 



mmmmmmMmtmm'90^ 



X I 



zm" z" 

n 



? + I and lefs than J^M^^ .^ 

2 ZwKpq-hn^i-hn "^ 



multiplied by the 3 terms 2 H — ~4 x ^^ 

?+2 



«+i 1 1 '-2 m'' z"^ 

5v '■» <» /^ 



where nf'y K, ^ 

and H ftand for the quantities already explained. 

An 



[ 404 ] 



An APPENDIX 

CONTAINING 

An Application of the foregoing Rules to fome parti- 
cular Cafes. 

np H E firft rule gives a diredt and perfed Iblution 
-■• in all cafes ; and the two following rules are 
only particular methods of approximating to the fo- 
lution given in the firft rule, when the labour of ap- 
plying it becomes too great. 

The firft rule may be ufed in all cafes where either 
p ov q are nothing or not large. The fecond rule 
may be ufed in all cafes where mz is lefs than VY; 
and the 3d in all cafes where m" z" is greater than 

I and lefs than - , if « is an even number and very 

large. If n is not large this laft rule cannot be much 
wanted, becaufe, m decreafing continually as n is 
diminifhed, the value of z may in this cafe be taken 
large, (and therefore a confidcrable interval had be- 
tween Z *- 2J and ±L 4- z^) and yet the operation be 

n n ' 

carried on by the 2d rulej ox mz not exceed x/y. 



But in order to (hew diftinftly and fully the nature 
of the prefent problem, and how far Mr. Bayes has 
carried the folution of it ; I fliall give the refult of 
this folution in a few cafes, beginning with the loweft 
and moft fimple. 

Let 



[ 405 ] 

Let us then firft fuppofe, of fuch an event as that 
called M in the eflay, or an event about the proba- 
bility of which, antecedently to trials, we know no- 
thing, that it has happened onccy and that it is en- 
quired what conclufion we may draw from hence 
with refped: to the probability of it's happening on a 
fecond trial. 

The anfwer is that there would be an odds of three 
to one for fomewhat more than an even chance that 
it would happen on a fecond trial. 

For in this cafe, and in all others where q is 



H*«PltM««aM««MMl • 



pj^l pj^l 



nothing, the expreflion n\^i %1L —a; 



or X * — a:^ "^ gives the folution, as will appear 
from confidering the firft rule. Put therefore m this 
expreflion J+7 = 2, X = i and a; = 4. and it will be 
I —T|* or 1.5 which fhews the chaoce there is that 
the probability of an event that has happened once 
lies fomewhere between i and 4. ; or (which is the 
fame) tlie odds that it is fomewhat more than an 
even chance that it will happen on a fecond trial *. 

In the fame manner it will appear that if the event 
has happened twice, the odds now mentioned will be 
feven to one ; if thrice, fifteen to one 5 and in gene- 
ral, if the event has happened p times, there will be 
an odds of 2/^ + » — i to one, for more than an equal 
chance that it will happen on further trials. 

Again, fuppofe all I know of an event to be that 
it has happened ten times without failing, and the 

* There c^Hj, I fuppofe, be no reafon for obferving that on 
this ful*je<ft unity is always made to fland for certainty, and -• 
for an isven chance. 

Vol. LIII. G g g enquiry 



[ 4.06 ] 

enquiry to be what reafon we fliall have to think we 
are right if we guefs that the probability of it's hap- 
pening in a fingle trial lies fomewhere between -i.^ 
and -*, or that the ratio of the caufes of it's happen- 
ing to thofe of it's failure is fome ratio between that 
of fixteen to one and two to one. ^. , 

Here/-|- i =z=: ii, X==4|. and x=z^ and X 
— • x^+^ =411" — H" == .501-2 &c. The anfwer 
therefore is, that we (hall have very nearly an equal 
chance for being right. 

In this manner we may determine in any cafe what 
conclufion we ought to draw from a given number 
of experiments which are unoppofed by contrary 
experiments. Every one fees in general that there is 
reafon to expecfl an event with more orlefs confidence 
according to the greater or lefs number of times in 
which, under given circumftances, it has happened 
without failing ; but we here fee exaftly what this 
reafon is, on what principles it is founded, and how 
we ought to regulate our expedations. 

But it will be proper to dwell longer on this 

head» 

Suppofe a folid or die of whofe number of fides 

and conftitution we know nothing ; and that we are 
to judge of thefe from experiments made in 
throwing it 

In this cafe, it fhould be obferved, that it would 
be in the higheft degree improbable that the folid 
fhould, in the firfl trial, turn any one fide v^hich could 
be afTigned before hand ; becaufe it would be known 
that fome fide it mufl turn, and that there was an in- 
finity of other fides, or fides otherv^ife marked, which 
it was equally likely that it fhould turn. The firfl 
A throw 



I 407 ] 

throw only (hews that it has the fide then thrown, 
without giving any reafon to think that it has it any 
one number of times rather than any other. It wilJ 
appear, therefore, that after the firft throw and not 
before, we fhould be in the circumftances required 
by the conditions of the prefent problem, and that 
the whole efFed of this throw would be to bring 
us into thefe circumftances. That is : the turning 
the fide firft thrown in any fubfequent fingle trial 
would be an event about the probability or improba- 
bility of which we could form no judgment, and 
of which we fhould know no more than that it 
lay fomewhere between nothing and certainty. With 
the fecond trial then our calculations muft begin; 
and if in that trial the fuppofed ibiid turns again the 
fame iide, there will arife the probability of three 
to one that it has more of that fort of fides than of 
all others ^ or (which comes to the fame) that there 
is fomewhat in its conftitution difpofing it to turn that 
fide ofteneft : And this probability will increafe, in 
the manner already explained, with the number of 
times in which that fide has been thrown without 
failing. It fhould not, however, be imagined that any 
number of fuch experiments can give fufficient reafon 
for thinking that it would never turn any other fide. 
For, fuppofe it has turned the fame fide in every 
trial a million of times. In thefe circumftances there 
would be an improbability that it had lefs than 
1.400,000 more of thefe fides than all others; but 
there would alfo be an improbability that it had above 
1.600,000 times more. The chance for the latter is 
expreffed by 4.4.^^4. raifed to the millioneth power 
fubftraded from unity, which is equal to .4647 &c.and 

G g g 2 the 



[ 408 ] 

the chance for the former is equal to i4^4^-o4i rzKci 
to the fame power, or to .4895; which, being both lefs 
than an equal chance, proves what I have faid. But 
though it would be thus improbable that it had ahve 
1.600,000 times more or kjs than 1400,000 liines 
more of thefe fides than of all others, it by no means 
follows that we have any reafon for judging that the 
true proportion in this cafe lies fomewhere between 
that of 1,600,000 to one and 1400,000 to one. 
For he that will take the pains to make the calcula- 
tion will find that there is nearly the probability ex- 
preflfed by .527, or but little more than an equal 
chance, that it lies fomewhere between that of 
600,000 to one and three millions to one- It may 
deferve to be added, that it is more probable that this 
proportion lies fomewhere between that of 900,000 
to I and 1.900,000 to i than between any other 
two proportions whofe antecedents are to one another 
as 900,000 to 1.900,000, and confequents unity. 

I have made thefe obfervations chiefly becaufe they 
arc all ftridly applicable to the events and appear- 
ances of nature. Antecedently to all experience, it 
would be improbable as infinite to one, that any par- 
ticular event, beforehand imagined, fhould follow 
the application of anyone natural objed to another ^ 
becaufe there would be an equal chance for any one of 
an infinity of other events. But if we had once feeii 
any particular effed:s, as the burning: of wood on 
putting it into fire,, ol the falling of a^ftone on de- 
taching it from all contiguous objefts , then the con^- 
clufions to be drawn from any number of fubfequent 
events of the fame kind would be to be determined 
ill the f^me manner with the eonclufions juft mcn^ 
tioned relating to the conftitution of the folid I have 

fuppofed 



[ 409 ] 

fuppofed. In other words. The firft experi- 
ment fuppofed to be ever made on any natural obje<5t 
would only inform us of one event that may follow a 
particular change in the circumftances of thofe objedts ; 
3ut it would not fuggeft to us any ideas of uniformity 
in nature, or give us the leaft reafon to apprehend 
that it was, in that inftance or in any other, regular ra- 
ther than irregular in its operations. But if the fame 
ey^nt has followed without interruption in any one 
or more fubfequent experiments, then fbme degree 
of uniformity will be obferved ; reafon will be given 
to exped the fame fuccefs in further experiments, and 
the calculations direded by the folution of this pro- 
blem may be made, ; ^ ^ 
One example here it will not be amrfs to give. 
Let us imagine to ourfelves the cafe of a perfon juft 
brought forth into this, world and left to colled from 
his obfervation of the order and courfe of events what 
powers and caufes take place in it. The Sun would, 
probably, be the firft objed that would engage his atten- 
tion; but after lofing it the firft night he would be en- 
tirelyignoran t whether he fliould ever fee it again. He 
would therefore be in the condtion of a perfon making a 
firft experiment about an event entirely unknown to 
him. But let him fee a fccond appearance or one 
return of the Sun, and an expedation would be raifed 
in him of a fecond return, and he might know that 
there was an odds of 3 to i ioxfome probability of this. 
This odds would increafe, as before reprefented, with 
the number of returns to which he was witnefs. 
But no finite number of returns would be fufficient 
to produce abfolute or phyfical certainty. For let it 
he fuppofed that he has feen it return at regular and 
ftated intervals a million of times. The conclufions 
5 this 



[ 410 ] 

this would warrant would be fuch as follow ' '■■"' ■ ■ ■ ^ 

There would be the odds of the millioneth power 
of 2y to one, that it was likely that it would return again 
at the end of the ufual interval. There would be the 
probability expreffed by .5352, that the odds for this 
was not ^r^^^^r than i. 600,000 to i ; And the pro- 
bability expreffed by .5105, that it was not kfs than 
1.400,000 to I. 

It fhould be carefully remembered that thefe de- 
dudions fuppofe a previous total ignorance of nature. 
After having obferved for fome time the courie of 
events it would be found that the operations of nature 
are in general regular, and that the powers and laws 
which prevail in it are ftable and parmanent. The 
confideration of this will caufe one or a few experi- 
ments often to produce a much ftronger expedation of 
fuccefs in further experiments than would otherwife 
have been reafonable j juft as the frequent obfervation 
that things of a fort are difpofed together in any place 
would lead us to conclude, upon difcovering there 
any objed of a particular fort, that there are laid up 
with it many others of the fame fort. It is obvious 
that this, fo far from contradiding the foregoing de- 
dudions, is only one particular cafe to which they are 

to be applied. 

What has been faid feems fufEcient to Ihew us 
what concluiions to draw from uniform experience. 
It demonftrates, particularly, that inftead of proving 
that events will always happen agreeably to it, there 
will be always reafon againft this conclufion. In other 
words, where the courfe of nature has been the moft 
conftant, we can have only reafon to reckon upon a 
recurrency of events proportioned to the degree of 

this 



[ 411 ] 

this conftancyj but we can have no reafon for thin Sl- 
ing that there are no caufes in nature which will ever 
inrerfere with the operations of the caufes from which 
this conftancy is derived, or no circumftances of the 
world in which it will fail. And if this is true> fup- 
poling our only data derived from experience, we fhall 
find additional reafon for thinking thus if we ap- 
ply other principles, or have recourfe to fuch conifi- 
derations as reafon, independently of experience, can 
fuggeft. 

But I have gone further than I intended here ^ and 
it is time to turn our thoughts to another branch of 
this fubje£t: I mean, to cafes where an experiment 
has fometimes fucceeded and fometimes failed. 

Here, again, in order to be as plain and explicit 
as poflible, it will be proper to put the following 
cafe, which is the eafieft and fimpleft I can think 
o£ 

Let us then imagine a perfon prefent at the drawing: 
of a lottery, who fncwsUin'g of its fchctne or of 
the proportion of Blanks to Prizes in it. Let it further 
be fuppofed, that he is obliged to infer this from the 
number of blanks he hears drawn compared with the 
number of prizes ^^ and that it is enquired what con- 
clufions in thefe circumftances he may reafonably 
make. 

Let him firft htzt ten blanks drawn and one^nzc^ 
and let it be enquired what chance he will have for be- 
ing right if he guefles that the proportion of Hanh to 
prizes in the lottery lies fome where between the pro- 
portions of 9 to I and 1 1 to i . 
^ Here taking X = ^4. x= A./=ro, q= 1, n^i i, 
E=; II, the required chance, according to the firft: 

rule. 



[ 412 ] 

rule, is tt+ I X E into the difference between 



— _ 



/>-f I 



9^ 



P + 2 



and 



/»+! 



p+2 



X 



p+i 



qx 



12 X II 



^^+2 



^+1 



/>+2 



rrv' III" 

JO » — . 121 



12 
II 



1. 

10 



ti 



1 
lO 



12. 



.07699 



12 



II 



12 



&c. There would therefore be an odds of about 923 
1076, or nearly 12 to i againfl his being right. Had 
he gueffed only in general that there were lefs than 
9 blanks to a prize, there would have been a proba- 
bility of his being right equal to .6589, or the odds 
of 65 to 34. 

Again, fuppofe that he has heard 20 blanh drawn 
and z prizes y what chance will he have for being 
right if he makes the fame guefs ? 

Here X and x being the fame, we have «= 22, 
fz=.2o^ f = ^j £ = 231, and the required chance 

"^^-fl p + 2 /»+3 

equal to «i+i x E x X - j'X + j'XJ'-i xX 



'p + 1 p + 2 



P + 3 



p+i 



X 



P + 2 p+3 

qx -j-^X^'-lX^ 



= .10843 &c. 

^+1 p + % 2 ^ + 3 

He will, therefore, have a better chance for being 
right than in the former inftance, the odds againft 
him now being 892 to 108 or about 9 to i. But 
Ihould he only guefs in general, as before, that there 
were lefs than 9 blanks to a prize, his chance for be- 
ing right will be worfe ; for inftead of .6589 or an 
odds of near two to one^ it will be .584, or an odds 
of 584 to 415. 

Suppofe, 



[ 413 ] 

Suppofe, further, that he has heard 40 bknh 
drawn and 4 prizes \ what will the before -mention- 
ed chances be ? 

The anfwer here is .1525, for the former of thefe 
chances; and .527, for the latter. There will, there- 
fore, now be an odds of only 54. to i againft the 
proportion of blanks to prizes lying between 9 to i 
and II to I ; and but little more than an equal chance 
that it is lefs than 9 to i. 

Once more. Suppofe he has heard 100 blanh 
drawn and i o prizes. 

The anfwer here may flill be found by the firft 
mle; and the chance for a proportion of blanks to 
prizes /g/S than 9 to i will be .44109, and for a pro- 
portion jr^^/^r than II to I .3082. It would there- 
fore be likely that there were not fewer than 9 or 
more than 1 1 blanks to a prize. But at the fame time 
it will remain unlikely * that the true proportion 
fhould lie between 9 to i and 1 1 to i, the chance 
for this being .2506 &c. There will therefore be 
jftill an odds of near 3 to i againft this- 

From thefe calculations it appears that, in the clr- 
cumftances I have fuppofed, the chance for beinff 
right in gueffing the proportion of blanks to prizes to 
be nearly the fame with that of the number of blanks 

* I fuppofe no attentive perfan will find any difficulty In this. 
It is only faying that, fuppofing the interval between nothing 
and certainty divided into a hundred equal chances, there will be 
44 of them for a lefs proportion of blanks to prizes than 9 to i, 
31 for a greater than 11 to i, and 25 for fome proportion be- 
tween 9 to I and II to 1 5 in which it is obvious that, though 
one of thefe fuppofitions muft be true, yet, having each of them 
more chances againft them than for them, they arc all feparately 
unlikely. 

Vol. LIII. H h h drawn 



[ 414 ] 

drawn in a given time to the number of prizes drawn, 
is continually increafing as thefe numbers increafe j 
and that therefore, when they are confiderably large, 
this conciufion may be looked upon as morally cer- 
tain. By parity of reafon, it follows univerfally, with 
refpeft to every event about which a great number 
of experiments has been made, that the caufes of its 
happening bear the fame proportion to the caufes of 
its failing, with the number of happenings to the 
number of failures; and that, if an event whofe 
caufes are fuppofed to be known, happens oftener or 
feldomer than is agreeable to this conciufion, there 
will be reafon to believe that there are fome unknown 
caufes which difturb the operations of the known 
ones. With rcfpedl, therefore, particularly to the 
courfe of events in nature, it appears, that there is 
demonftrative evidence to prove that they are derived 
from permanent caufes, or laws originally eftabliftied 
in the conftitution of nature in order to produce that 
order of events which we obferve, and not from any 
of the powers of chance *. This is juft as evident 
as it would be, in the cafe I have infifted on, that the 
reafon of drawing lo times more blanks th^n prizes 
in millions of trials, was, that there were in the wheel 
about fo many more blanks than prizes. 

But to proceed a little further in the demonftration 
of this point. 

We have feen that fuppofing a perfon, ignorant of 
the whole fcheme of a lottery, (hould be led to con- 
jecture, from hearing loo blamsaad lo prizes drawn^ 

* See Mr, De Moivre's Dodrine of Chances, pag. 250. 

that 



[ 415 3 

that the proportion of blanks to prizes in the lottery 
was fbme where between 9 iio i and xi to 1/ the 
chance for his being right would be ♦2506 &c. Let 
now enquire what this chance would be in fome 
h^her cafes. 

Let it be fuppofed that blanks haye been drjiwn 
1000 times, and prizes 106 times In iroo trials. 

In this cafe the powers of 3£ gnd x rife fo high, 

and the number of terms in the two ferides X 
^ qX &c« and a; — §^x &c. become 

to obtain the anfwer by the rf rule. Tb ncceffiiy, 
therefbrc, to ba^ rccourfe to thefecond rule. But 
in order to make ufe of it, the inmryal Tbetween X 
and ^ muft be a little altered. 44 *- -^ is ^4.^ audi 
rilerefbre the interval between 4^"^ - -^^ and 44 

TT*3r 



will be nearly the fame with the interval be-* 
tweeh 4w ^^ iiy oi^ly fomewhat larger. If then 
wc make the queftion to be ; what chance there 
wbiild be (fiippofinsr no more known thm that blgnka 
pe been dSwn^ooo times and prizes lao tlmm 
in iioo trials) that the probability of drawing a 
blank in a fingle trial would lie fomewhere between 
^^.-^^4:^ and 44 4" TTTir w^ ^^^^ have a queftion 
of thfe lame kind with the preceding queftiotfiSi and 
deviate but little from the limits dBgned in fiiembi 
The aniwer, according to the fccond eul^ m &^ 

litis chance is greater than i^ z Ea^ &^ + zl^ a?Tf 



n 



Hhh 2 and 



[ 4i6 ] 



a X 



W» n i l »i mmmmmmmm ■ ■ » i m j i m — — i— iM 

and lefs than 1--2 E ^^^ d^ ^-a E^^ ^f, E being n-^ 1 



n n 

.3 ~3 



^ .JtLXxBa'^ p^ Xmz^ + — f x — -- &c, 

V « ^ 3 2» 5 

By making here 1000 z=zp iooz=:q iioo=:« 
being the ratio whofe hyperbolic logarithm is -^'^ X 

I I I II I I.I I I I tf«.^ 

n p q 360^ n^ p f 1260 «^ ^^ f^ 

and K the ratio of the quadrantal arc to radius 5 the 
former of thefe expreffions will be found to be .7953, 
and the latter .9405 &c. The chance enquired after, 
therefore, is greater than .7953, and lefs than .9405* 
That is; there will be an odds for being right in gueff- 
ing that the proportion of blanks to prizes lies marly 
between 9 to i and 1 1 to i, (or exa^Iy between 9 to 
I and 1 1 1 1 to 99) which is greater than 4 to i, 
and lefs than 16 to i. 

Suppofe, again, that no more is known than that 
Hanks have been drawn 10,000 times and prizes 1000 
times in iiooo trials ^ what will the chance now 
mentioned be? 

Here the fecond as well as the firft rule becomes 
ufelefs, the value of mz being fo great as to r ender 

it fcarcely poffible to calculate diredly the feries Hi^ 

^^+2l!x^^&c. The third rule, therefore, 
3 3t» 5 

muft be Ujfed ; and the information it gives us is, that 

the required chance is greater than •97421, or more 

than an odds of 40 to i. 

By 



[ 4^7 ] 

By calculations fimilar to thefe may be determined 
univerfally, what expedtations are warranted by any 
experiments, according to the different number of 
times in which they have fucceeded and failed; or 
what fhould be thought of the probability that any 
particular caufe in nature, with which we have any 
acquaintance, will or will not, in anyfingle trial, 
produce an efFed that has been conjoined with it. 

Moft perfons, probably, might expert that the 
chances in the fpecimen I have given would have been 
greater than I have found them. But this only fhews 
how liable we are to error when we judge on this 
fubjedt independently of calculation. One thing, 
however, fhould be remembered here; and that 
is, the narrownefs of the interval between ^% and 
44, or between 4.4 + ^4^ and 4.^ — ^4.^. Had 
this interval been taken a little larger, there would 
have been a confiderable difference in the ref^Us of 
the calculations. Thus had it been taken double, or 
z =:^V> ^' would have been found in the fqurth in- 
ftance that inftead of odds againft there were oddi 
for being right in judging that the probability of draw- 
ing a blank in a lingle trial lies between 44 -|- ^ and 



Xr TT* 



The foregoing calculations further fhew us the 
ufes^ and delfts of the rules laid down in the effay. 
*Tis evident that the two laft rules do not give us 
the required chances within fuch narrow limits as 
could be wiihed; But here again it fhould be confi- 
dered, that thefe limits become narrower aqd narrow- 
er as §^ is taken larger in reipeft of py arid when^ 
and y are equal, the exadl folution is given it% all cafes 
by the fecond rule. Thefe two rules therefore afford 

a direction 



I ^,1 8 ] 

tDc^f|ti%t4fent that may be of tehfideiv 
llbte^jie^Hij&mi ijsrleift {liall difcover a better ajji 

^MAstinti 3iC©Ttfig> vdie of the two feries's in the 

5{fi^|jlilvJ|g|iiirt^(p|ftoIirecommends the ibltition iil 

\hh m^ 1^ A#l¥t H^'compleat in thofe cafes whePi 
Mstrflliisli p^j mbfttcwaiited, and where Mr. De 
Moi^^#it)ltobifofc lise inverfe problem can give 
lilllet«s|J rf^3#f ^<^^^I«I mean, in all cafes where ei- 
tM4 pi0|o^i<a«eco|i|ai cohfiderable magnitude. Iwi 
^iMSfft ^fes,8idl ts«Henboth / and ^ are very Confider- 
ab%,«ft igpotf diffictttd to perceive the truth of whafc 
U^^^tem^e 4mmkMmbd, or that thefe is reafoii tti 
feillevl4a geffeidl te?#ife chances for the happening 
y%^5,e\^ftlWs:yj«h#Hdhances for its failOre in the 
feife ratjb w'i^h t^it b£|& to q. But we fhall be greatly 
l3beei?ed#i3Wd|i«ige:ihlijis manner when either /bf' 
IfMfM^^ii AhdaiiscftJiH' fuch cafes the D^jifa are nol 
^fBtlrfflpfoftifeSverfie^ekaa: probability of ian events 
yet lil^J(^r|lgfleelible3io be? able to find the limits be* 
tyeen3^(M;h:iitfe fei^abfe to think it muft lie, and 
aVi tb le I^feWdeietriline the precife degree of aflent 
kM^,Mmmmp<hrm^xiMms or aflertions relating 
to them. 

•Xf^Sirii^teyirW^tfekk have found out a methocl of confi- 
iteril)lgiii|)Sb«i%?»fena|^bxiniation in the ad and ad rules by 

diSS#Pl%ill»f rtf eg^feFi^on 1+2 E tf^ ^y + 2 £ i7^ ^ comes 




ffii||Ktt#f rM^ii i¥(^li&rbe given. 






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